Analyses of transient acoustic radiation are often encountered in engineering applications because most vibrating structures are subject to impulsive or transient force excitations. In many cases, the transient excitations are unspecified and therefore the structural vibration responses are unknown, which makes the analysis of resulting acoustic field difficult. While there are available numerous research papers on predicting transient acoustic radiation from a finite object subject to the given excitations or time-dependent boundary conditions, literature on reconstruction of transient acoustic radiation is almost nonexistent. The closest work is the use of spherical near-field scanning to predict far-field acoustic radiation in the time domain. This scarceness is caused by a lack of an effective methodology to reconstruct transient acoustic radiation based on a limited number of acoustic pressure measurements.
One way of dealing with transient acoustic radiation from an arbitrary object is via the Kirchhoff-Helmholtz integral theory, which correlates the field acoustic pressure to the surface acoustic pressure and normal surface velocity. These surface acoustic quantities are governed by the Kirchhoff-Helmholtz integral equation and must be solved before the field acoustic pressure can be calculated. For an arbitrary surface, this Kirchhoff-Helmholtz integral theory can only be implemented numerically through boundary element method (BEM) by discretizing the surface into elements and interpolating the acoustic quantities on the discrete nodes. Since the acoustic quantities on these nodes have different emission times for an observer in a fixed coordinate system, the integrals become time dependent. Accordingly, one must discretize the integrals in both time and spatial domains, thus making numerical computations extremely time consuming. Needless to say, it will be even more difficult, if possible at all, to apply the Kirchhoff-Helmholtz integral theory to transient reconstruction.
An alternative in transient NAH is to reconstruct the acoustic quantities in the frequency domain first, and then take an inverse Fourier transform to retrieve time domain signals. Transient acoustic radiation from a vibrating object has been reconstructed in this manner. It has been demonstrated that the transient acoustic field can be reconstructed in the frequency domain using the Helmholtz equation least squares (HELS) method, and then taking an inverse Fourier transform to recover the time history of the normal surface velocity response. The infinite integral in the inverse Fourier transform is calculated by direct numerical integration for a vibrating sphere in a free field. Needless to say, the numerical computations involved in this reconstruction process are huge. Therefore, a direct calculation of an inverse Fourier transform to reconstruct transient acoustic radiation is not advisable.
The temporal acoustic field has been reconstructed via the so-called non-stationary spatial transformation of sound field (NS-STSF). NS-STSF is based on the time domain holography (TDH) that processes the acoustic pressures measured by a two-dimensional microphone array with the neighboring microphones separated by half wavelength. TDH “can be seen as a sequence of snapshots of instantaneous pressure over the array area, the time separation between subsequent snapshots being equal to the sampling interval in A/D conversion. Similarly, the output of TDH is a time sequence of snapshots of a selected acoustic quantity in a calculation plane parallel to the measurement plane.” So what one sees are acoustic pressure and intensity distributions in the frequency domain at some fixed instants over the measurement time period. As such, NS-STSF is not transient NAH in the sense that it cannot provide a three-dimensional image of an acoustic signal traveling in space and time. Moreover, NS-STSF is very restrictive: it only allows reconstruction of the acoustic quantities on the same grids as the measurement grids on a plane parallel to the measurement plane in a free field.